Usage of Mutex in c#

The problem here is that all your callers are using a different mutex; you need the locking object to be shared, usually by making it a field. For example, and switching to a simpler lock metaphor:

private readonly object syncLock = new object();
public void ThreadSafeMethod() {
    lock(syncLock) {
        /* critical code */
    }
}

or using the mutex:

private readonly Mutex m = new Mutex();
public void ThreadSafeMethod() {
    m.WaitOne();
    try {
        /* critical code */
    } finally {
        m.ReleaseMutex();
    }
}